Genericity on submanifolds and application to Universal hitting time statistics

We investigate Birkhoff genericity on submanifolds of homogeneous space $X=S{L}_d(\text{bR})⋉({\text{bR}}^d{)}^k\mathrm{/}S{L}_d(\text{bZ})⋉({\text{bZ}}^d{)}^k$, where $d\ge 2$ and $k\ge 1$ are fixed integers. The submanifolds we consider are parameterized by unstable horospherical subgroup $U$ of a diagonal flow $a_t$ in $S{L}_d(\text{bR})$. As long as the intersection of the submanifold with any affine rational subspace has Lebesgue measure zero, we show that the trajectory of $a_t$ along Lebesgue almost every point on the submanifold gets equidistributed on $X$. This generalizes the previous work of Fr\k{a}czek, Shi and Ulcigrai in \cite{Shi_Ulcigrai_Genericity_on_curves_2018}. Following the scheme developed by Dettmann, Marklof and Str"{o}mbergsson in \cite{Marklof_Universal_hitting_time_2017}, we then deduce an application to universal hitting time statistics for integrable flows.